Dimension of an affine subspace

Alicia Y. Tsai; Authors: Laurent EI Ghaoui; Contributors: Phan Hoang, Tony Tin, Ha To Tam, Nguyen Van Kep, Le Dinh Nam, Juliette Decugis, Nguyen Quoc Cuong.; Giuseppe C. Calafiore

Dimension of an affine subspace

The set ${\bf L}$ in $\mathbb{R}^3$ defined by the linear equations

$\begin{align*} x_1 - 13x_2 + 4x_3 &= 2, \\ 3x_2 - x_3 &= 9 \end{align*}$

is an affine subspace of dimension $1$ . The corresponding linear subspace is defined by the linear equations obtained from the above by setting the constant terms to zero:

$\begin{align*} x_1 - 13x_2 + 4x_3 &= 0, \\ 3x_2 - x_3 &= 0 \end{align*}$

We can solve for $x_3$ and get $x_1 = x_2, \; x_3 = 3x_2$ . We obtain a representation of the linear subspace as the set of vectors $x \in \mathbb{R}^3$ that have the form

$\begin{align*} x_1 &:= \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix} t, \end{align*}$

for some scalar $t=x_2$ . Hence the linear subspace is the span of the vector $u:=(1, 1, 3)$ , and is of dimension $1$ .

We obtain a representation of the original affine set by finding a particular solution $x^0$ , by setting say $x_2 = 0$ and solving for $x_1, x_3$ . We obtain

$\begin{align*} x^0 &:= \begin{pmatrix} 38 \\ 0 \\ -9 \end{pmatrix}. \end{align*}$

The affine subspace ${\bf L}$ is thus the line $x^0 + {\bf span}(u)$ , where $x^0, u$ are defined above.

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